Averages summarise data with a single representative value. GCSE requires you to calculate mean, median, and mode, and to know when each is most useful.
Core Concepts
Mean
Affected by outliers.
Median
The middle value when data is ordered.
For values: position = .
For even : average of the two middle values.
Mode
The most common value. Can have no mode, one mode, or multiple modes.
Range
Measures spread.
Mean from a Frequency Table
Estimated Mean from Grouped Data
Use midpoints:
Worked Example: Example 1
Problem
Data: 3, 5, 7, 7, 8, 10. Mean = . Median = . Mode = 7. Range = 7.
Solution
Worked Example: Grouped Data
Problem
| Score | Freq | Midpoint | f×m |
|---|---|---|---|
| 0-10 | 5 | 5 | 25 |
| 10-20 | 8 | 15 | 120 |
| 20-30 | 7 | 25 | 175 |
Estimated mean = .
Solution
When to Use Which
- Mean: when data is symmetric with no outliers.
- Median: when data is skewed or has outliers.
- Mode: for categorical data.
Practice Problems
- Find mean, median, mode: 4, 6, 6, 8, 11.
- Estimated mean from a grouped frequency table.
- The mean of 5 numbers is 8. Four are 5, 7, 9, 10. Find the fifth.
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Key Takeaways
Mean = sum ÷ count (affected by outliers).
Median = middle value (robust to outliers).
Mode = most common.
Grouped data: use midpoints for estimated mean.